Question: In an \[\Delta ABC\] , if \[\angle A=\pi /2\] , then \[{{\cos }^{2}}B+{{\cos }^{2}}C\] equals A.\[...

Question

In an $\Delta ABC$ , if $\angle A=\pi /2$ , then ${{\cos }^{2}}B+{{\cos }^{2}}C$ equals
A. $-2$
B. $-1$
C. $1$
D. $0$

Answer 1

Solution

To solve this problem, first use the property of triangles and try to find out the sum of the rest of two angles of a triangle and then substitute that value in the given equation and then use trigonometric identity and trigonometric properties to simplify it and you will get your required answer.

Complete answer: Trigonometry can be defined as a study of the relationship of angles, lengths and heights.
. There are total six types of different functions in trigonometry: Sine $(\sin )$ , Cosine $(\cos )$ , Secant $(\sec )$ , Cosecant $(\cos ec)$ , Tangent $(\tan )$ and Cotangent $(\cot )$ . Basically these six types of trigonometric functions define the relationship between the different sides of a right angle triangle.
Sine is defined as the ratio of opposite side to the hypotenuse
Cosine is defined as the ratio of adjacent side to the hypotenuse
Tangent can be defined as the ratio of opposite side to the adjacent side
Cotangent is the reciprocal of the tangent. So it is the ratio of adjacent side to the opposite side.
Cosecant is the reciprocal of sine. So it can be defined as the ratio of hypotenuse to the opposite side.
Secant is the reciprocal of the cosine. So it can be defined as the ratio of hypotenuse to adjacent side.
Now, according to the given question:
We have given,
In $\Delta ABC$ , $\angle A=\dfrac{\pi }{2}$

We know that the sum of all the angles in a triangle is equals ${{180}^{o}}$
So, in $\Delta ABC$
$\angle A+\angle B+\angle C=\pi$
$\dfrac{\pi }{2}+\angle B+\angle C=\pi$
$\angle B+\angle C=\pi -\dfrac{\pi }{2}$
$\angle B+\angle C=\dfrac{\pi }{2}$
$\angle B=\dfrac{\pi }{2}-\angle C$
Now, ${{\cos }^{2}}B+{{\cos }^{2}}C$
$\Rightarrow {{\cos }^{2}}\left( \dfrac{\pi }{2}-C \right)+{{\cos }^{2}}C$
$\Rightarrow {{\sin }^{2}}C+{{\cos }^{2}}C$
$\Rightarrow 1$
Hence, ${{\cos }^{2}}B+{{\cos }^{2}}C=1$
Therefore, from all the above options $C$ is the correct option.

Note:
Triangles are three sided polygons. There are several types of triangles such as: Equilateral triangle, isosceles triangle and scalene triangle etc. In a triangle, when three altitudes meet a common point then that point is called the orthocenter of the triangle.